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We give an improvement of the Caratheodory theorem for strong convexity (ball convexity) in $mathbb R^n$, reducing the Caratheodory number to $n$ in several cases; and show that the Caratheodory number cannot be smaller than $n$ for an arbitrary gauge body $K$. We also give an improved topological criterion for one convex body to be a Minkowski summand of another.
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In 1940, Luis Santalo proved a Helly-type theorem for line transversals to boxes in R^d. An analysis of his proof reveals a convexity structure for ascending lines in R^d that is isomorphic to the ordinary notion of convexity in a convex subset of R^{2d-2}. This isomorphism is through a Cremona transformation on the Grassmannian of lines in P^d, which enables a precise description of the convex hull and affine span of up to d ascending lines: the lines in such an affine span turn out to be the rulings of certain classical determinantal varieties. Finally, we relate Cremona convexity to a new convexity structure that we call frame convexity, which extends to arbitrary-dimensional flats.
Inspired by the classical Riemannian systolic inequality of Gromov we present a combinatorial analogue providing a lower bound on the number of vertices of a simplicial complex in terms of its edge-path systole. Similarly to the Riemannian case, where the inequality holds under a topological assumption of ``essentiality, our proofs rely on a combinatorial analogue of that assumption. Under a stronger assumption, expressed in terms of cohomology cup-length, we improve our results quantitatively. We also illustrate our methods in the continuous setting, generalizing and improving quantitatively the Minkowski principle of Balacheff and Karam; a corollary of this result is the extension of the Guth--Nakamura cup-length systolic bound from manifolds to complexes.
A set of vertices $S$ of a graph $G$ is a (geodesic)convex set, if $S$ contains all the vertices belonging to any shortest path connecting between two vertices of $S$. The cardinality of maximum proper convex set of $G$ is called the convexity number, con$(G)$ of $G$. The complementary prism $Gbar{G}$ of $G$ is obtained from the disjoint union of $G$ and its complement $bar{G}$ by adding the edges of a perfect matching between them. In this work, we examine the convex sets of the complementary prism of a tree and derive formulas for the convexity numbers of the complementary prisms of all trees.
The average kissing number of $mathbb{R}^n$ is the supremum of the average degrees of contact graphs of packings of finitely many balls (of any radii) in $mathbb{R}^n$. We provide an upper bound for the average kissing number based on semidefinite programming that improves previous bounds in dimensions $3, ldots, 9$. A very simple upper bound for the average kissing number is twice the kissing number; in dimensions $6, ldots, 9$ our new bound is the first to improve on this simple upper bound.
We give a simple proof of T. Stehlings result, that in any normal tiling of the plane with convex polygons with number of sides not less than six, all tiles except the finite number are hexagons.
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